SOME RELATIONSHIPS AMONG INTERNAL CONSISTENCY, REPRODUCIBILITY, AND HOMOGENEITY

Abstract

One property of such tests which has received a great deal of attention from psychometricians is the of the test. Internal has been characterized in a variety of ways by different writers. Kuder and Richardson (1937) provided an early formulation which was subsequently developed and extended by Cronbach (1951). This approach is derived from classical reliability theory. The indices based upon this conception are commonly called measures of internal consistency. Guttman (1944) defined internal structure based upon a scaling model. His formulation determines the degree to which item responses conform to a unidimensional structure. The indices based upon his conception are called measures of reproducibility. Loevinger (1947) has proposed a variation upon Guttman's approach which compares observed item covariances with corresponding maximum possible values. Indices based upon Loevinger's approach are usually called measures of homogeneity. Measures of internal consistency, reproducibility, and homogeneity clearly are conceptually interrelated. Indices based upon each of the three formulations start with the same basic data-the N X K matrix of 0-1 item scores for each of N respondents on each of K items. Cronbach (1951) and White and Saltz (1957) have noted some of the general relationships among the various indices that are commonly employed. However, formal proofs and illustrations of certain relationships of interest have not been given. It is the purpose of this paper (1) to present a proof of the relationship of the maximum and minimum values of certain indices of reproducibility, homogeneity, and internal consistency, and (2) to illustrate how maximum and minimum values of the most widely employed index of internal consistency are related to test difficulty and the nature of the distribution of total test scores.